NCTM San Diego 2010 Fractions, Ratios and Patterns: Helping Elementary Students Get Ready for Algebra New Teacher Workshop Eomailani Bettercourt, University of Hawai' i, Manoa Joe Zilliox, University of Hawai' i, Manoa Emphasis on Process standards: Problem solving, Communication, Connections, Reasoning, and Relationships Fraction as: A part-whole relationship. A place on a number line. An operator in computation. Opening Question 2 7 is a representation that tells me something. What does it say to you? Activity 1: Folding Paper Strips Using the colored strips and making all creases along the short dimension, fold to show the following fractions: gray is one-whole; fold the yellow into halves, blue into thirds, green into fourths, pink into sixths, and purple into eighths Use the portions you created to 2 Show 7 in three different ways. On a number line 3 in exactly in the middle between 2 and 4. Is 7 exactly in the middle between —, and 7 ? Use the idea from above to decide where 7 would be if it made with the same strips. Comparing fractions without common denominators. Use the strip and logical reasoning to explain why 7 is greater or less than 7 ? (Note: what kind of language is useful here). Now use similar reasoning without calculations to compare the following. 3 5 7 7 13 17 a. q and 7 b. 73 and 77 c. 77 and < g What portions match — ? What names do we give to these portions. Combiner + 7 - make the portion. Name the portion. Fractions, Ratios and Patterns: Helping Elementary Students Get Ready for Algebra - New Teacher Workshop NCTM San Diego 2010 Activity 2: Fractions on a number line Label 15 post-its with the following fraction numerals: i2' 3i 3 'n4' 4i' 6 i' 6 i' 8 'i 8 'i8 'H i i i i i 8 ' 12 ' 12 ' 12 ' 12 Suppose the left end of the log paper strip (adding machine tape) is 0 on the number line and the right end is the number 1. Using only estimation (no folding), place each of the labeled post-its on the number line where it belongs relative the 0 and the 1. When all are placed stand back and reconsider the location of the post-its, moving those you think are misplaced. Compare your marks to the other group at your table. At this point you may or may not want to make changes. Once the placements are decided, the paper strip can be marked with the fraction name in pencil (because the post-its tend to fall off) to record the locations. To check their estimates, take a second paper strip and carefully fold it into 24 equal portions. Label this strip with the same fraction names used on the first strip. If the folding is done carefully these labels should be more accurate than the estimates. A good estimate should be within a post-it width on either side of the fold line. Activity 3: Patterns and Generalizations in Fraction Sums and Differences Calculate: (a)j + 3 = ©3+4= (c)t+7 = What patterns or connections do you see between the addends and the sums? Can you generalize this patter in words and symbols. Repeat for the following differences (a)j- 3 = (b)"-4=(c)4-5 = Do the above patterns work if the denominators are not consecutive integers (2 and 7 instead of2and3)? Repeat each of the above by using 2 in numerator of both fractions. How does the pattern or generalization change? Activity 4: How Does It Grow - (see separate handout) Look at the "row houses" in the first pattern. 1. What is the ratio of the number of triangles to the number of squares? Squares to blocks? Blocks to triangles? 2. If I have 15 triangles, how many squares will I need? How many houses will be in the last set? 3. If I have 72 squares how many triangles will I need? How many houses in the last set? 4. If I have 234 blocks and they make row houses without any left over, how many of the blocks were squares and how many were triangles? How many houses made up the last set? Fractions, Ratios and Patterns: Helping Elementary Students Get Ready for Algebra - New Teacher Workshop NCTM San Diego 2010 Activity 5: Ratios with Pattern Blocks One of the vendors in the NCTM exhibit area is giving away small samples bags of their pattern blocks. Each bag is exactly the same and contains four triangles, three trapezoids, and five squares. Mr and Ms Freebee collected many sample bags so they can do projects with their students. a. If their first project requires 12 triangles, when they open the sample bags to get the triangles how many squares will there be? How many trapezoids? b. If the second project requires a total of 48 blocks, how many sample bags will they need and how many of each kind of block will they have when they open the bags? c. A final project requires 64 quadrilaterals. How many of these will be squares and how many will be trapezoids? Activity 6: Pattern Block Similarity 1. Begin with one square from the set of pattern blocks. Using only orange squares make another larger square using as few small squares as possible. Still using only orange squares make the next largest square using as few small squares as possible. There are now three figures composed of squares. For each of the three figures record the length of a side, perimeter, and area. What patterns can be found in these number sequences? What will be the next number in each sequence? Why? 2. Leave the squares intact on the table. Repeat this activity but instead of orange squares use just triangles, or just blue rhombi, or red trapezoids. So use trapezoids to make the next larger trapezoid. How are the number patterns the same or different as they were for the square? What does this have to do with ratios? 3. Now take a single square in your hand. Stand over the larger square made with four small squares. Place the single square in your hand between one of your eyes and the four squares on the table as you look down on them. At some point in your line of vision between your eye and the four squares, the single square should appear to cover the four squares exactly. Hold that position. Measure the distance between the eye and the small square and then measure the distance between the eye and the larger square on the table. How do the measures compare? Activity 7: Function Machine See separate handout Fractions, Ratios and Patterns: Helping Elementary Students Get Ready for Algebra - New Teacher Workshop NCTM San Diego 2010 Function Machines Input \ L Function Rule 7 \ Output Building Towers The idea of a function machine is that you begin with a starting input value as the first number in a sequence and apply the "rule" to produce an output. That output becomes the second number in the sequence. To get additional numbers in the sequence the output is repeatedly returned to the machine as the next input. Example: Use 1 as the starting number and with a rule of "add 1" we would get the following sequence: A. 1, 2, 3,4, 5, The numbers represent the number of blocks in a tower. Find the following sequences using the starting number and the rule. Stop after 5 towers are in your sequence. B. Starting number is 3, rule is "add 1". C. Starting number is 1, rule is "add 2". D. Starting number is 1, rule is "double the input". E. Starting number is 1, rule is "add n, where n is 1, then 2, then 3,...". Sketch the five on grid paper. Use the card stock to connect the towers. What does the slant in the card stock indicate? Name A AA AAA The pattern continues. 1. Draw the next figure. 2. When there are six triangles, how many squares are there? 3. When there are five triangles, how many shapes (squares and triangles) are there in all? 4. Can there be seven squares in a figure? Why? This pattern continues. 5. Draw the next figure. 6. When there are ten squares, how many triangles are there? 7. How do you know? 8. When there are fourteen triangles, how many squares are there? 9. How do you know? Copyright © 2001 by the National Council of Teachers of Mathematics, Inc. www.iiclm.org. All rights reserved. Navigating through Algebra in Prekindergarten-Grade 2 Name Navigating through Problem Solving and Reasoning in Grade 4 Copyright © 2005 by the National Council of Teachers of Mathematics, Inc. www'.nctm.or<r. All rights reserved.
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